What is this book about?

The theme of this book is the allocation of n objects (or elements) into g categories (or classes), discussed from several viewpoints. This approach can be traced back to the early work of 24-year-old Ludwig Boltzmann in his first attempt to derive Maxwell's distribution of velocity for a perfect gas in probabilistic terms.

Chapter 2 explains how to describe the state of affairs in which for every object listed ‘alphabetically’ or in a sampling order, its category is given. We can consider the descriptions of Chapter 2 as facts (taking place or not), and events as propositions (true or not) about facts (taking place or not). Not everything in the world is known, and what remains is a set of possibilities. For this reason, in Chapter 3, we show how events can be probabilized and we present the basic probability axioms and their consequences. In Chapter 4, the results of the previous two chapters are rephrased in the powerful language of random variables and stochastic processes.

Even if the problem of allocating n objects into g categories may seem trivial, it turns out that many important problems in statistical physics and some problems in economics and finance can be formulated and solved using the methods described in Chapters 2, 3 and 4. Indeed, the allocation problem is far from trivial. In fact, in the language of the logical approach to probability, traced back to Johnson and, mainly, to Carnap, the individual descriptions and the statistical descriptions are an essential tool to represent possible worlds.